Fitting Highly Non-linear Data

Hi all,
At the moment I am trying to find a best fit equation to radial velocity data vs. time of a planet HD17156b. The paper with the Authors fit parameters (I am trying to mimic the fit) has the arXiv ref number of :0704.1191v2
From this paper I extract their data, which is on the final few pages and take note of their fit parameters.
I use the relevant equations to make a continuous curve using there fit parameters. They say in the paper they achieve a fit with a normalised Chi squared value = 1.17
When I plot the data, and the continuous curve I note that the fit is definitely not close to 1. I believe this is because of the nature of the equations used are highly non-linear and the error in their fit parameters are throwing off the continuous curve.
When I try to find a best fit using the Newton-Raphson method in conjunction with the Levenberg-Marquardt Algorithm I use their best fit parameters as my initial starting point. Assuming surely this would find a good fit to the data. But I achieve a horrible fit. Anybody have any ideas that could help me achieve a better fit similar to the original Authors fit?
Below is a plot of the original Data and the Authors best fit parameters leading to the continuous curve on the plot.

It sounds like you are using a 'downhill' maximum likelyhood (aka 'best fit') method. In the case of a real parameter space, with real noisy data and, as you say, non-linear equations, it is quite possible that the parameter space (i.e. Chi squared as a function of the fit parameters) is not a smooth function with a single global minima. You may be stuck in a local minima, like rolling a marble down a rocky slope and the marble gets stuck in a hole before it reaches the bottom.

Maximum likelyhood fitting is IMHO a bad idea. It works sometimes but you never know in advance when it will and when it will not. A better approach is to map the parameter space and see how the solution behaves. The most general approach is a Markov Chain Monte Carlo random walk through the parameter space.

Rather than losing it in the re-telling, have a look at this paper. It is a marvelous case study into the pitfalls of maximum likelyhood fitting of extrasolar planet data. It shows that the maximum likelyhood fit of Tinney et al. 2003 for the same data was actually the least likely of 3 possible orbits. I would suggest reading this paper, and others by the same author, and implementing the kind of fitting algorithm they use.

Hmm, actually I think the paper I described is a different one, though by the same author. I think it is Gregory 2005a that is mentioned in the second paragraph of the paper I linked to. It might be better to read that one first since the one I linked I think is an extension. In any case that paper should still be very useful, but the actual algorithm is probably described in more detail elsewhere. Play follow the citations and you should find it!